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Configurations and decoupling : a few problems in Euclidean harmonic analysis Yang, Tongou

Abstract

In this thesis, we study two topics in Euclidean harmonic analysis. The first one is the configurations contained in fractal-like sets in the Euclidean space. The other is decoupling for various geometric objects in the Euclidean space. In the study of Euclidean configurations, we first discuss the background, address their subtleties and do a simple survey on this subject. Then we proceed to the proof of my main result, which demonstrates the topological property of a set containing a similar copy of sequences converging to zero. In the study of decoupling, we first formulate a general decoupling inequality and discuss some general upper and lower bound estimates Then we move on to decoupling for manifolds in Euclidean space, and in particular curves in the plane. We then state a classical result by Bourgain and Demeter and use it to prove a decoupling inequality that works uniformly for all polynomials up to a certain degree, generalising an earlier result of Biswas et al. in the plane.

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Attribution-NonCommercial-NoDerivatives 4.0 International

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